Non-Perturbative QCD at Finite Temperature · Non-Perturbative QCD at Finite Temperature Pok Man Lo University of Pittsburgh Jlab Hugs student talk, 6-20-2008 Pok Man Lo (UPitt) Non-Perturbative

Non-Perturbative QCD at Finite Temperature

Pok Man Lo

University of Pittsburgh

Jlab Hugs student talk, 6-20-2008

Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature 6-20-2008 1 / 39

personal information

email: [email protected]

institution: University of Pittsburgh

advisor: Eric Swanson


outline

1 Non-perturbative QCD at zero temperatureIntroduction to non-perturbative physicsTools for studying non-perturbative QCD

DiagrammaticsSchwinger Dyson Equation

Illustration: Gap Equation Revisited

2 Finite Temperature Field TheorySurvey of basic formalism of Finite Temperature of Field TheoryFinite Temperature Gap Equation


Outline






Introduction

The progress of theoretical physics

The search for exact solution:

Newtonian physics: 3-body problem was insoluble

QED: 2-body and 1-body problem was insoluble

now: 0-body, namely, the vacuum is insoluble

No body is too many!!


Introduction








Introduction








Introduction








Introduction








Introduction








Introduction

Difficutly in calculating arbitarily complicated diagram

Impossible to sum all the diagrams


Introduction

Difficutly in calculating arbitarily complicated diagram

Impossible to sum all the diagrams


Introduction

Perturbation: expansion in α

is not useful if α is not small

cannot explain non-perturbative phenomena:e.g bound state formation and spontaneous symmetry breaking

classical mechanics example: mass attached to a spring: A sin(√

km t)


Introduction





km t)


Introduction





km t)


Introduction





km t)


Outline






illustration of the failure of perturbation

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10

...an innocent looking function

using Taylor expansion, we write

f (x) = A0 + A1x + A2x2 + ...

A0,A1,A2, ... are all strightly 0!



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10

...an innocent looking functionusing Taylor expansion, we write

f (x) = A0 + A1x + A2x2 + ...




0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10

...an innocent looking functionusing Taylor expansion, we write

f (x) = A0 + A1x + A2x2 + ...




differential equation...

f (x) + x3f ′ = 0

f (x) = exp−1x2

any other method will work, other than perturbation!




f (x) + x3f ′ = 0

f (x) = exp−1x2





f (x) + x3f ′ = 0

f (x) = exp−1x2





f (x) + x3f ′ = 0

f (x) = exp−1x2



Different Philosophy

Perturbative Vs Non-perturbative:

perturbative method: sum all diagrams up to certain order in α

non-perturbative method: sum a certain class of diagram to all order

in some sense, non-perturbative method includes the whole perturbationmethod as the latter is just a classification of diagram by the couplingconstant αapproximation 6= perturbation!QFT contains more than just the S-matrix and perturbation!


















in some sense, non-perturbative method includes the whole perturbationmethod as the latter is just a classification of diagram by the couplingconstant α

approximation 6= perturbation!QFT contains more than just the S-matrix and perturbation!






in some sense, non-perturbative method includes the whole perturbationmethod as the latter is just a classification of diagram by the couplingconstant αapproximation 6= perturbation!

QFT contains more than just the S-matrix and perturbation!








Outline






motivation

perturbation does not help as we want to sum diagrams to all order in α

new tools other than perturbation!

in some sense, we need exact relations among diagrams!

two particularly useful ones:

Diagrammatics

Schwinger Dyson Equation


method of partial sum

summing a particular class of diagrams to all order

to be explicit, let’s consider the diagrams for constructing a propagator


method of partial sum

summing a particular class of diagrams to all order

to be explicit, let’s consider the diagrams for constructing a propagator


diagrammatics of propagator

in general, to construct the propagator, we need to sum...



in general, we cannot sum them all!

but let’s focus on the following class of diagram:











the above procedure should be compared with the summing of geometricseries:

1

1− x= 1 + x + x2 + x3 + ...



Two comments:

inspired by the series, we know we can do better:



Two comments:




Two comments:




⇐⇒ 6p −m



⇐⇒ 6p −m



⇐⇒ dynamical mass



⇐⇒ dynamical mass



an exact relations among n-point functions

basic idea: ∫dx

d

dxf (x)− > 0

Z =

∫DψDψDGe i(S+

Rηψ+ψη+...)/

∫DψDψDGe i(S)

using the fact ∫DψDψDG (

δ

δψe i(S+

Rηψ+ψη+...)) = 0

Z = e iW




basic idea: ∫dx

d

dxf (x)− > 0

Z =

∫DψDψDGe i(S+

Rηψ+ψη+...)/

∫DψDψDGe i(S)


δ

δψe i(S+

Rηψ+ψη+...)) = 0

Z = e iW




basic idea: ∫dx

d

dxf (x)− > 0

Z =

∫DψDψDGe i(S+

Rηψ+ψη+...)/

∫DψDψDGe i(S)


δ

δψe i(S+

Rηψ+ψη+...)) = 0

Z = e iW




basic idea: ∫dx

d

dxf (x)− > 0

Z =

∫DψDψDGe i(S+

Rηψ+ψη+...)/

∫DψDψDGe i(S)


δ

δψe i(S+

Rηψ+ψη+...)) = 0

Z = e iW



as an illustration, for the Hamiltonian:

H =

∫d3xψ†~xγ

0[−i~γ · ∇+ m]ψ~x − G

∫d3xd3yV~x~yψ

†~xT

aψ~xψ†~yT aψ~y


(iγ · ∂x −m)δ2Wδηxδηy

+ 2G

∫d4zV γ0T ai

δ2Wδηxδηz

γ0T a δ2Wδηzδηy

= δxy


Outline






Gap Equation Revisited

to illustrate the main idea above, we look at the gap equation

consider the Hamiltonian:

H =

∫d3xψ†~xγ

0[−i~γ · ∇+ m]ψ~x − G

∫d3xd3yV~x~yψ

†~xT

aψ~xψ†~yT aψ~y

for the contact case: V → δ(3)

this corresonds to the case where the quark exchange an instantaneousgluon locally



to illustrate the main idea above, we look at the gap equation

consider the Hamiltonian:

H =

∫d3xψ†~xγ

0[−i~γ · ∇+ m]ψ~x − G

∫d3xd3yV~x~yψ

†~xT

aψ~xψ†~yT aψ~y

for the contact case: V → δ(3)

this corresonds to the case where the quark exchange an instantaneousgluon locally



the gap equation tell you how the quark is dressed according to theHamiltonian

namely, the dynamical mass generated:

before showing you the answer, we need to perform our final dressing...


the final dressing

in fact we miss some diagram...


the final dressing


the final dressing



the gap equation for the general case:

Generalized Gap Equation

M(~k) = m − GTr [TT ]

Nc

∫d3k ′

(2π)3V~k ′−~k [

M(~k ′)

E~k ′− k̂ ′ · k̂ |

~k ′|M(~k)

E~k ′ |~k|]

M(~k) in general is a function of ~k

it dictates how the mass is dynamically generated


Gap Equation Revisitedfor the contact case, we have...

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 1 1.5 2 2.5 3 3.5 4

M

current mass m

Dynamical Mass Generation

|G| > G_critical|G| < G_critical


Outline






motivation

FTFT is needed to study the physics of QGP, deconfinement and chiralrestoration

when calculating observables in QFT, we only calculate the vacuumexpectation value

at finite temperature, excited states start to contribute, the interestingquantity should be the thermal average of the observables

we expect nE~kto enter QFT

partition function dictates the equilibrium Finite Temperature QFT


Outline






basics of FTFTthe partition function:

Z = Tr [e−βH ] =

∫dq < q|e−βH |q >

observables are given by

O = Tr [e−βHO] =<< q|O|q >working in imaginary time τ = it

a corresponding path integral representation of the partition function, withPeriodic/Antiperiodic boundary condition

the boundary condition motivates the use of Matsubara Green’s function:

∫dk0 → 1

βΣωn


Outline






Gap equation in Finite Temperature

Generalized Gap Equation

M(~k) = m− GTr [TT ]

Nc

∫d3k ′

(2π)3V~k ′−~k [

M(~k ′)

E~k ′− k̂ ′ · k̂ |

~k ′|M(~k)

E~k ′ |~k |](1− 2nE~k′

)

with nE~k= 1

eβE~k +1


summary

non-perturbative physics with e−1x2

summation of a class of diagram with 11−x

Schwinger Dyson Equation with∫

dx ddx f (x)

Finite Temperature Field Theory with Tr [e−βH ]


thank you


Documents

Non-Perturbative QCD at Finite Temperature · Non-Perturbative QCD at Finite Temperature Pok Man Lo University of Pittsburgh Jlab Hugs student talk, 6-20-2008 Pok Man Lo (UPitt) Non-Perturbative